A sequence of groups is just a list of groups with homomorphisms going down the list: . We’ll use to refer to the homomorphism from to . We say that a sequence is “exact” if the image of is the kernel of for each .
What does this mean? First off, if we start with an element of and hit it with we get something in . Since this is the same as , if we now apply we go to the identity element of . That is, the composition is always the trivial homomorphism sending all of to the identity in .
On the other hand, if we have an element in the kernel of it’s also in the image of . This means that if an element gets sent to the identity in the next group, it must have come from an element in the previous group.
So why should we care? Well, a number of different things can be very nicely said with exact sequences. If we write for the group containing only one element, we can set up a sequence: . What does it mean for this sequence to be exact? Well there’s only one homomorphism from to any group, and its image is just the identity element in . So the kernel of is trivial — is a monomorphism.
Now let’s flip the diagram over to . There’s only one homomorphism possible from any group to , and its kernel is the whole domain. This means that the image of has to be all of — is an epimorphism.
Let’s put these two together to get a sequence . Exactness at means that is a monomorphism, which we can think of as describing a copy of sitting inside . Exactness at means that is an epimorphism. What does exactness at mean? The image of is that copy of , which has to also be the kernel of . That is, is (isomorphic to) . We call any sequence of this form a “short exact sequence”.
Remember that the First Isomorphism Theorem tells us that we can factor any homomorphism into an epimorphism from the domain onto a quotient, followed by a monomorphism putting that quotient into the codomain. We can use that here to weave any exact sequence out of short exact sequences. Here is the (really cool) diagram:
Each of the diagonal lines is a short exact sequence, and as it says each (nontrivial) group off the main line is the image of one of the homomorphisms on the line and the kernel of the next.
We can also write an exact sequence . This just says that the homomorphism between and is an isomorphism. It’s really nice when this shows up in the middle of a longer exact sequence. If we can show that and are both trivial the sequence looks like , so and are immediately isomorphic.
Another way exact sequences show up is in describing the structure of a group. We know that every group is a quotient of a free group. That is, there is some free group so that is exact. Then the kernel of this projection is another group, so it’s the quotient of another free group . Now the sequence is exact. This is the presentation of by generators and relations. But the homomorphism from to might have a nontrivial kernel — there might be relations between the relations. In that case we can describe those relations as the quotent of another free group : is exact. We can keep going like this to construct an exact sequence called a “free resolution of “. It’s particularly nice if the process terminates at some point, giving a sequence . A free resolution of a group that has only finitely many terms gives a lot of information about the structure of .